Each harmonic was described by four Fourier coefficients, two each for
the x -and y -axes, generating a total of 4ncoefficients labeled an, bn,
cn and dn, where n is the number
of harmonics. They combine to describe the repeated elements in a
sinusoidal waveform, (aj , bj ) and (cj , dj )
were the four Fourier coefficients defining each harmonic (j th
order); k corresponds to the maximum number of harmonics used for the
Fourier decomposition and T was equal to the perimeter of the outline.
An incremental number of harmonics in Fourier series converges on
detailed information on the morphology of analyzed form, and numerically
describes distortions and complexity of forms (from the original circle)
described by the Fourier series when no harmonics are present. In this
manner, Elliptical Fourier Analysis (EFA) is able to resolve and
describe complex 2-Dimensional bounded outlines and also spatial
orientation relation to reference planes (Ferrario et al., 1996),
allometric changes and shape (Procrustes coordinates) can be assessed
when plotted against CS. The size normalization procedure consisted of a
recalculation of the outlines using the same value of the enclosed area
for all specimens.
Measurement error was estimated from a Procrustes ANOVA by considering
individual as the main source of variation and residuals representing
variation in digitized replicates as a second source of error (Cocilovo
et al. 2012). To remove differences due to specimen orientation and
position during data collection, and to separate size and shape
components; landmark configurations are first scaled to the same size,
centered at their origin and rotated to minimize the distances among the
corresponding landmarks (Generalized Procrustes Analysis or GPA) (Rohlf,
2010). GPA superimposes specimen landmark configurations by translating
them to a common origin, scaling them to unit centroid size (the square
root of the sum of squared distances of all landmarks to the centroid of
the object), and rotating them according to a best-fit criterion
Figure 2a and b Showing dorsal, ventral and direct caudal views
of the small African Pangolin skull (P. tricuspis ) and landmarks
assessed 1. Posthion in the midline of the asal bone, 2. Suture of the
nasal bone in the midline rostral limit of the frontal bone, 3 Parietal
suture in the midline at the rostral limit of the parietal bone, 4 the
Inter-parietal bone suture at the caudal limit of the parietal in the
midline, 5 Dorsal limit of the foramen magnum in the midline of the
occipital bone, 6 and 7 Lateral boundaries of the nasal bone sutures in
the maxilla, 8 and 9 Lateral limits of the parietal bone sutures at the
temporal bones, 10 and 11 Lateral limits of the Inter-parietal bone
sutures with the occipital bone. The ventral view landmarks includes; 1
the Posthion in the ventral view, 2 the ventral midline boundary of the
sphenoid, 3 Ventral rim of the foramen magnum in the ventral midline, 4
and 5 Ventral tips of the zygoma on both sides, 6 and 7 Ventro-lateral
limits of the temporal bone on both sides, 8 and 9 The lateral
boundaries of the occipital condyles in ventral view (b) the caudal view
of foramen magnum outline
Size normalization of descriptor coefficients of foramen outlines was
achieved by the ellipse of the first harmonic described for closed
shapes (Kuhl and Giardina, 1982) to be invariant of size employing CHc
2-NEF SHAPE Version 1.3. Major axis length/2, Minor axis length/2,
orientation of the major axis and phase angle (θ = 1/2 arct.2
(a1b1 +
c1d1)/a12+ c12 –
b12 –
d12) corresponding to the position
elliptical first point are the four descriptors of each harmonic. A
stepwise Elliptical Fourier reconstruction of foramen magnum outlines
was done employing incremental number of harmonics. Size and peculiar
anisotropic characteristics were determined for each harmonic (Major
axis length/2 x Minor axis length/2 and Major axis/Minor axis
respectively) (Buck, 1962).
Descriptor coefficients were analyzed with ‘PrinComp’ based on
Variance-Covariance matrix of normalized coefficients. It was also
noted that such coefficients with small variance and covariation values
do not significantly explain morphological variations and are now
calculated and used in principal components derivation bearing all
edge/contour shapes information in the first 14 harmonics as explained
by Rholf and Archie, (1984). Summary statistics showing mean ± SD
employing PAST (Hammer et al., 2013) for foramen magnum shape outline
descriptors was performed. An allometric (shape vs size) based
discriminant function analysis was evaluated with incremental Fourier
descriptor harmonics followed by use of inverse Fourier transform to
obtain graphical representation at 0.05 Bonferroni post-test level of
significance.